# A bead glides frictionless on a wire that has the shape of a cycloid 
# g is the gravitational acceleration 9,81 m/s²
# a is the radius of the rolling circle (see Bronstein/Semendjajew p. 91)
# equation of motion:
#   mu'' = - g/4a * mu, with mu = sin(phi/2) and phi a parameter of the cycloid 

alias coefficient.1 g/4a
coefficient.2 (+1) -> mu0'
coefficient.3 (-1) -> -mu0
alias coefficient.4 4ax
alias coefficient.5 4ay # same as 4ax

iintegrate mu'' -> -mu'
   IC: mu0'
iintegrate -mu' -> mu
   IC: -mu0
invert mu -> -mu
coefficient.g/4a (-mu) -> -g/4a*mu
assign -g/4a*mu -> mu''

# the following is for displaying the cycloid in x-y space
# calculating x (NB: this includes some unacceptable approximations)
coefficient.4ax (mu) -> 4a*mu
output(4a*mu) -> out.x

# calculating y
multiply mu, mu -> mu^2
coefficient.4ay (mu^2) -> 4ay*mu^2
isum 4a*mu^2 -> -2a*mu^2     # just serves to devide by 2 because we need 2a instead of 4a
  /2
invert -2a*mu^2 -> 2a*mu^2 
output(2a*mu^2) -> out.y 

# display mu, so the sinus
output(mu) -> out.z